Definition
In calculus, a limit describes the value that a function $f(x)$ approaches as the input $x$ gets arbitrarily close to a specific point $a$. Formally, the limit of $f(x)$ as $x$ approaches $a$ is written as:
$$ lim_{x \to a} f(x) $$
If this limit exists, it tells us the behavior of $f(x)$ near $x = a$, regardless of the function's value at $a$.
Worked Example
Problem: Find
$$ lim_{x \to 2} \frac{x^2 - 4}{x - 2} $$
Step 1: Substitute $x = 2$ directly:
$$ \frac{2^2 - 4}{2 - 2} = \frac{4 - 4}{0} = \frac{0}{0} $$
This is an indeterminate form, so we need to simplify.
Step 2: Factor the numerator:
$$ x^2 - 4 = (x - 2)(x + 2) $$
So,
$$ \frac{x^2 - 4}{x - 2} = \frac{(x - 2)(x + 2)}{x - 2} $$
Step 3: Cancel $(x - 2)$ (for $x ne 2$):
$$ = x + 2 $$
Step 4: Now take the limit as $x \to 2$:
$$ lim_{x \to 2} (x + 2) = 2 + 2 = 4 $$
Takeaways
- Limits describe the behavior of functions as inputs approach a specific value.
- If direct substitution gives an indeterminate form (like $\frac{0}{0}$), try algebraic simplification.
- Understanding limits is foundational for concepts like continuity and derivatives in calculus.